L11a463

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_8,14,9,13 X_22,8,13,7 X_20,15,21,16 X_16,6,17,5 X_18,12,19,11 X_12,18,5,17 X_10,20,11,19 X_2,9,3,10 X_4,22,1,21
Gauss code	{1, -10, 2, -11}, {6, -1, 4, -3, 10, -9, 7, -8}, {3, -2, 5, -6, 8, -7, 9, -5, 11, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 -2 v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + v 3 u 2 -3 v 2 u 2 + 4 v u 2 - v 3 w u 2 + 3 v 2 w u 2 -4 v w u 2 + 2 w u 2 -2 u 2 -2 v 3 u + 4 v 2 u -3 v u + 2 v 3 w u -4 v 2 w u + 3 v w u - w u + u + v 3 -2 v 2 + v + 2 v 2 w - v w$ (db)
Jones polynomial	$q 9 -3 q 8 + 7 q 7 -9 q 6 + 15 q 5 -17 q 4 + 17 q 3 -15 q 2 + 12 q -7 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 2 z 4 a -4 -2 z 4 a -6 - z 4 + z 2 a -2 + 2 z 2 a -4 -4 z 2 a -6 + z 2 a -8 - z 2 + a -4 -3 a -6 + a -8 + 1 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 5 z 9 a -3 + 10 z 9 a -5 + 5 z 9 a -7 + 6 z 8 a -2 + 4 z 8 a -4 + 4 z 8 a -6 + 6 z 8 a -8 + 6 z 7 a -1 -8 z 7 a -3 -30 z 7 a -5 -13 z 7 a -7 + 3 z 7 a -9 -7 z 6 a -2 -18 z 6 a -4 -30 z 6 a -6 -22 z 6 a -8 + z 6 a -10 + 4 z 6 + a z 5 -9 z 5 a -1 + 9 z 5 a -3 + 32 z 5 a -5 + 5 z 5 a -7 -8 z 5 a -9 - z 4 a -2 + 24 z 4 a -4 + 49 z 4 a -6 + 28 z 4 a -8 -3 z 4 a -10 -7 z 4 - a z 3 + z 3 a -1 -7 z 3 a -3 + 11 z 3 a -7 + 2 z 3 a -9 -12 z 2 a -4 -31 z 2 a -6 -20 z 2 a -8 + z 2 a -10 + 2 z 2 -11 z a -5 -11 z a -7 + 5 a -4 + 13 a -6 + 8 a -8 + 1 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of L11a463. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 7$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}$	${\mathbb Z}$